講義名 英語理学講義（数学２）（ Science in English (Mathematics2) ）
開講学期 後学期 単位数 1--0--0
担当 Andrei Pajitnov 東京工業大学理学研究流動機構客員教授 （フランス・ナント大学教授）
Circle-valued Morse theory and Novikov homology
1) Novikov complex for circle-valued Morse functions.
2) Witten-type de Rham framework for Novikov inequalities.
3) Morse-Novikov theory for 3-knots
4) Applications to symplectic topology.
Classical Morse theory establishes a relation between the number
of critical points of a real-valued function on a manifold and the
topology of the manifold.
Circle-valued Morse theory is a branch of the Morse theory.
It originated from a problem in hydrodynamics studied by
S. Novikov in the early 1980s.
Nowadays it is a constantly growing field of geometry with
applications and connections to many geometric problems
such as Arnold's conjectures in symplectic topology,
fibrations of manifolds over the circle, dynamical zeta functions,
and the theory of 3-dimensional knots and links.
Our course will start with recollections on the classical Morse
theory (gradient flows, Morse complex, Morse inequalities).
Then we proceed to the circle-valued Morse theory and
construct the Novikov complex (the generalization of the Morse
complex to the case of circle-valued functions).
This chain complex is generated over the Laurent series ring
by the critical points of the circle-valued function.
E. Witten showed in the beginning of 1980s that the classical Morse
theory can be reformulated in the framework of the de Rham theory.
We will explain his work, and its generalization to the Morse-Novikov
theory, which leads to a relation between the Novikov homology
and the homology with local coefficients.
A natural application of the circle-valued Morse theory is to the
topology of knots. If a knot K in the 3-dimensional sphere S is not
fibred, then any circle-valued function on S-K has critical points.
The minimal number of these critical points is an invariant of K,
called the Morse-Novikov number MN(K).
We will show how to give computable lower bounds for MN(K)
with the help of the Novikov homology, and relate it to the tunnel
number of the knot.
In the last part of the course we will explain how the Novikov ring
appears in the Floer's work on the Arnold conjecture concerning
the closed orbits of periodic Hamiltonians, and discuss some recent
developments in this direction.